On a Countable-Point Nonlinear Boundary-Value Problem on a Semiaxis for Ordinary Differential Equations in the Space of Bounded Numerical Sequences

Teplins'kyi, Yu. V.; Nedokis, V. A.

doi:10.1023/b:nono.0000028589.05624.4f

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“…Let D βϕh (in [5], this set was denoted by D β ) denote the subset of D each point x 0 of which is contained in the set D together with its β ϕh -neighborhood, where…”

Section: Boundary-value Problem On a Semiaxismentioning

confidence: 99%

“…was originated in [1][2][3][4], where a periodic boundary-value problem for first-order and second-order equations and a twopoint boundary-value problem for a nonlinear first-order equation with linear boundary condition were investigated. Countable-point boundary-value problems for ordinary nonlinear differential equations of normal form defined in the space M were studied in [5,6].The present paper is a logic continuation of [5,6]. We consider here a boundary-value problem with nonlinear boundary condition that contains an unbounded countable set of boundary moments on the positive semiaxis, whereas the equation and the boundary condition of the problem are defined in the space M and, moreover, the equation is unresolved with respect to the derivative.…”

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confidence: 99%

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confidence: 99%

“…The present paper is a logic continuation of [5,6]. We consider here a boundary-value problem with nonlinear boundary condition that contains an unbounded countable set of boundary moments on the positive semiaxis, whereas the equation and the boundary condition of the problem are defined in the space M and, moreover, the equation is unresolved with respect to the derivative.…”

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confidence: 99%

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Boundary-value problems for differential equations unresolved with respect to the derivative in the space of bounded number sequences

2007

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We study a nonlinear countable-point boundary-value problem for a differential equation unresolved with respect to the derivative. This equation and a nonlinear boundary condition are defined in the Banach space of bounded number sequences. We study the reducibility of the posed problem to a multipoint boundary-value problem in a finite-dimensional space.It is clear that the possibility of the constructive investigation of various boundary-value problems for ordinary differential equations substantially depends on the dimension of the space in which a boundary-value problem is considered. At present, boundary-value problems of various types are best studied in finite-dimensional spaces. The investigation of these problems in abstract Banach spaces was carried out in relatively few works, mainly devoted to periodic boundary-value problems. The study of boundary-value problems in the Banach space M of bounded sequences x = (x 1 , x 2 , x 3 , . . . ) of real numbers with norm x = sup i {|x i |, i = 1, 2, 3, . . . } was originated in [1][2][3][4], where a periodic boundary-value problem for first-order and second-order equations and a twopoint boundary-value problem for a nonlinear first-order equation with linear boundary condition were investigated. Countable-point boundary-value problems for ordinary nonlinear differential equations of normal form defined in the space M were studied in [5,6].The present paper is a logic continuation of [5,6]. We consider here a boundary-value problem with nonlinear boundary condition that contains an unbounded countable set of boundary moments on the positive semiaxis, whereas the equation and the boundary condition of the problem are defined in the space M and, moreover, the equation is unresolved with respect to the derivative. In the case where the set of boundary moments belongs to the finite segment [0, T ], the initial boundary-value problem is reduced to an analogous multipoint boundary-value problem in a finite-dimensional space. Note that a three-point boundary-value problem with linear boundary condition for an equation in a finite-dimensional space that is unresolved with respect to the derivative was considered in [7].

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“…Let D βϕh (in [5], this set was denoted by D β ) denote the subset of D each point x 0 of which is contained in the set D together with its β ϕh -neighborhood, where…”

Section: Boundary-value Problem On a Semiaxismentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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